Spin

Detailed Description

These objects are combined using a vertex to form a generalised current that can be used to compute the angular momentum components of the transition matrix.
In general, these objects are only described for decays of the type \( 1\to 2 \), with more complex structures constructed from such quasi two-body decays (see Isobar model), but more general amplitudes can also be defined.

Namespaces
namespace	AmpGen::Vertex
	Namespace that contains the base class for vertices, Vertex::Base, as well as the implementations of specific spin couplings and some helper functions such as the orbital operators.

Classes
struct	AmpGen::Vertex::Base
	Base class for all spin vertices. More...
struct	AmpGen::Vertex::S_SS_S
	\( S = S_1 S_2 \) More...
struct	AmpGen::Vertex::S_VV_S
	\( S = g_{\mu\nu} V_1^\mu V_2^{\nu} \) More...
struct	AmpGen::Vertex::S_VV_S1
	\( S = S_{\mu\nu} V_1^\mu V_2^{\nu} \) More...
struct	AmpGen::Vertex::S_VV_P
	\( S = \varepsilon_{\alpha\beta\mu\nu} P^{\alpha} L^{\beta} V_1^{\mu} V_2^{\nu} \) More...
struct	AmpGen::Vertex::S_VV_D
	\( S = L_{\mu\nu} V_1^\mu V_2^\nu \) More...
struct	AmpGen::Vertex::S_VS_P
	\( S = L_{\mu} V_1^{\mu} S_2 \) More...
struct	AmpGen::Vertex::S_TV_P
	\( S = L^{\mu} T_{\mu\nu} V^{\nu} \) More...
struct	AmpGen::Vertex::S_TV_D
	\( S = \varepsilon_{\mu\nu\alpha\beta} T^{\mu\gamma} L_{\gamma}^{\nu} P^{\alpha} V^{\beta} \) More...
struct	AmpGen::Vertex::S_TS_D
	\( S = T^{\mu\nu} L_{\mu\nu}\) More...
struct	AmpGen::Vertex::S_TT_S
	\( S = T_1^{\mu\nu} T_{2\mu\nu}\) More...
struct	AmpGen::Vertex::V_SS_P
	\( V^{\mu} = L^{\mu} S_1 S_2 \) More...
struct	AmpGen::Vertex::V_VS_S
	\( V^{\mu} = S^{\mu\nu} V_{1\nu} S_2 \) More...

Macros
#define	DECLARE_VERTEX(NAME)
	Macro to declare a vertex.

Functions
Tensor	AmpGen::Orbital_PWave (const Tensor &p, const Tensor &q)
	Helper function that computes the \(L=1\) orbital momentum operator.
Tensor	AmpGen::Orbital_DWave (const Tensor &p, const Tensor &q)
	Helper function that computes the \(L=2\) orbital momentum operator.
Tensor	AmpGen::Spin1Projector (const Tensor &p)
	Helper function that computes the projection operator onto a spin one state.
Tensor	AmpGen::Spin2Projector (const Tensor &p)
	Helper function that computes the projection operator onto a spin one state.
Tensor	AmpGen::Spin1hProjector (const Tensor &B)
	Helper function that projects a spinor.
double	AmpGen::CG (const double &j1, const double &m1, const double &j2, const double &m2, const double &J, const double &M)
	Calculates the Clebsch-Gordan coefficient for (j1 m1 j2 m2 | J M), the expansion coefficients in
TransformSequence	AmpGen::wickTransform (const Tensor &P, const Particle &p, const int &ve=1, DebugSymbols *db=nullptr)
	Generates a wick transform sequence that aligns tensor P (four-vector) to the +/- ve z-axis, then boosts to the rest frame.

Macro Definition Documentation

DECLARE_VERTEX

#define DECLARE_VERTEX

(

NAME

)

Value:

  struct NAME : public Base {                                                                                       \
    NAME(){ DEBUG("Constructing vertex");                    }                                                         \
    virtual Tensor operator()(const Tensor& P, const Tensor& Q, const Tensor& V1, const Tensor& V2, DebugSymbols* db = 0 ) override;  \
    static std::string _id;                                                                                            \
  }

Definition at line 26 of file Vertex.h.

Function Documentation

CG()

double AmpGen::CG	(	const double &	j1,
		const double &	m1,
		const double &	j2,
		const double &	m2,
		const double &	J,
		const double &	M )

Orbital_DWave()

Tensor AmpGen::Orbital_DWave	(	const Tensor &	p,
		const Tensor &	q )

Helper function that computes the \(L=2\) orbital momentum operator, which is given by

\[ L_{\mu\nu} = L_{\mu}L_{\nu} - \frac{L^2}{3} S_{\mu\nu}, \]

where \( L_{\mu} \) is the Orbital_PWave operator and \( S_{\mu\nu} \) is the Spin1Projector.

Orbital_PWave()

Tensor AmpGen::Orbital_PWave	(	const Tensor &	p,
		const Tensor &	q )

Helper function that computes the \(L=1\) orbital momentum operator, which is given by

\[ L_{\mu} = q_{\mu} - p_{\mu} \frac{p_{\nu}q^{\nu}}{ p^2 }, \]

where \( p \) is total momentum and \( q \) is the momentum difference between the two particles in the state.

Spin1hProjector()

Tensor AmpGen::Spin1hProjector

(

const Tensor &

B

)

Helper function that projects out a spin-half state

\[ S_{ab} = \frac{1}{2m}\left( {p\!\!\!/} + m I \right) \]

Spin1Projector()

Tensor AmpGen::Spin1Projector

(

const Tensor &

p

)

Helper function that projects some lorentz object onto the momentum \(p_\mu \) of state, and is given by

\[ S_{\mu\nu} = g_{\mu\nu} - \frac{p_{\mu}p_{\nu}}{p^2}. \]

Spin2Projector()

Tensor AmpGen::Spin2Projector

(

const Tensor &

p

)

Helper function that projects some lorentz object onto the momentum \(p_\mu \) of state, and is given by

\[ S_{\mu\nu\alpha\beta} = \frac{1}{2}\left( S_{\mu\alpha} S_{\nu\beta} + S_{\mu\beta}S_{\nu\alpha} \right) - \frac{1}{3} S_{\mu\nu} S_{\alpha\beta}, \]

where \( S_{\mu\nu} \) is the spin-one projection operator (see Spin1Projector).

wickTransform()

TransformSequence AmpGen::wickTransform	(	const Tensor &	P,
		const Particle &	p,
		const int &	ve = 1,
		DebugSymbols *	db = nullptr )

The mass may be seperately specified. The parameter ve specifies whether the initial Euler rotation is to the +/- z-axis. In the case where ve =-1, a second rotation is applied about the x-axis that aligns P to the +ve z-axis. This ensures that singly and doubly primed helicity frames remain orthonormal.